Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,9)(3,7,10,4)(5,8,14,16)(11,15,13,12)(17,18)(19,25)(20,28)(21,24)(22,26)(23,30)(27,31)(29,32)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(1,3,6,10)(2,7,9,4)(5,13,14,11)(8,12,16,15)(17,18,21,24)(19,22,26,25)(20,23,30,28)(27,31,29,32)(33,34,37,41)(35,39,43,38)(36,44,45,40)(42,46,48,47);(3,10)(4,7)(11,13)(12,15)(19,26)(22,25)(27,29)(31,32)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,16)(14,15)(17,19,21,26)(18,22,24,25)(20,29,30,27)(23,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,15)(12,13)(14,16)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,32)(26,29)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,45)(47,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,16)(10,13)(17,18)(19,22)(20,23)(21,24)(25,26)(27,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,44)(43,48)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8735039);